MAST10016 Exam formula sheet PDF

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MAST10016 Mathematics for Biomedicine Formula sheet Probability laws P (E 10 ) = 1  P (E1 )

Complement law

P (E1 \ E2 ) P (E2 ) P (E1 \ E2 ) = P (E1 ) · P (E2 )

Multiplication law for independent events

P (E1 [ E2 ) = P (E1 ) + P (E2 )

Addition law for mutually exclusive events

Conditional probability

P (E1 |E2 ) =

Discrete random variables E(X) =

X

kP (X = k)

k2D

Binomial distribution ✓ ◆ n k p (1  p)nk P (X = k) = k

✓ ◆ n n! = n Ck = k k !(n  k )!

where

E(X) = np

Var(X) = np(1  p)

X-linked genes The allele frequencies of alleles A, a of an X-linked gene satisfy the equations 1 1 ffA(n + 1) = ffA (n) + ffA (n  1) n1 2 2 A fm (n + 1) = f fA (n). The solution of equation (1) is ✓ ◆n 1 . ffA(n) = c1 + c2  2

(1)

Fisher-Haldane-Wright model Adult genotype frequencies: pAA(n) =

wAA f AA (n) w(n)

pAa(n) =

wAa f Aa (n) w(n)

paa(n) =

waaf aa(n) w(n)

w(n) = wAAf AA (n) + wAa f Aa (n) + waaf aa(n) Adult allele frequencies: pn+1 =

wAA (pn )2 + wAapn [1  pn ] wAA (pn )2 + 2wAapn [1  pn ] + waa [1  pn ]2

= p n + p n qn

hspn + s(1  h)qn p2n + 2(1  hs)pn qn + (1  s)qn2

where pn = pA (n), qn = pa(n) and

The equilibria of pn are

w bAa =

pn = 0

wAa = 1  hs wAA pn = 1

w baa =

waa =1s wAA

pn = P =

h1 2h  1

MAST10016 Formula sheet - page 1 of 2

if h 6=

1 2

Markov chains π(n) = [π0 (n) π1 (n) . . . π N (n)] = π(0)P n Wright-Fisher model ✓ ◆ N A (n) N A (n + 1) ⇠ Bi 2N, 2N N A (0) . 2N 1  p0 loge (1  p0 ). The mean time to fixation is t⇤ (p0 ) ' 4N p0 ✓ ◆   1 n The expected heterozygosity is E H(n) = H(0) 1  2N where H(n) = 2f A (n)f a(n) = E(f Aa(n + 1)).

The probability of fixation of A is Π= p0 =

Moran model The transition probabilities are given by 8 2   i N i 2 j = i >

: 0 otherwise The probability of fixation of A is Π= p0 = Enzyme kinetics dp Vmax s Under EA: = KD + s dt dp Vmax s Under QSSA: = KM + s dt dp Vmax sh Hill kinetics: = h dt KH + sh

N A (0) . N

where

KD =

k−1 k1

where

KM =

k−1 +k2 , k1

where

h  1, Vmax = k2 e0

, Vmax = k2 e0 Vmax = k2 e0

SIR model ds = βsi dt

di = βsi  γi dt

dr = γi dt

R0 = β/γ

End of formula sheet

MAST10016 Formula sheet - page 2 of 2...